The Mass-To-Light Function of Virialized Systems and the Relationship Between Their Optical and X-ray Properties

We compare the B-band luminosity function of virialized halos with the mass function predicted by the Press-Schechter theory in cold dark matter cosmogonies. We find that all cosmological models fail to match our results if a constant mass-to-light ratio is assumed. In order for these models to match the faint end of the luminosity function, a mass-to-light ratio decreasing with luminosity as $L^{-0.5\pm 0.06}$ is required. For a $\mathrm{\Lambda }$CDM model, the mass-to-light function has a minimum of $\sim 100h_{75}^{-1}$ in solar units in the $B$-band, corresponding to $\sim 25$ of the baryons in the form of stars, and this minimum occurs close to the luminosity of an $L^{\ast }$ galaxy. At the high-mass end, the $\mathrm{\Lambda }$CDM model requires a mass-to-light ratio increasing with luminosity as $L^{+0.5\pm 0.26}$. We also derive the halo occupation number, i.e. the number of galaxies brighter than ${\text{lgal}}^{\ast }$ hosted in a virialized system. We find that the halo occupation number scales non-linearly with the total mass of the system, $N\sbr{gal}(>\lgal^*) \propto m^{0.55\pm0.026}$orthe$\Lambda$CDM model. We find a break in the power-law slope of the X-ray-to-optical luminosity relation, independent of the cosmological model. This break occurs at a scale corresponding to poor groups. In the $\mathrm{\Lambda }$CDM model, the poor-group mass is also the scale at which the mass-to-light ratio of virialized systems begins to increase. This correspondence suggests a physical link between star formation and the X-ray properties of halos, possibly due to preheating by supernovae or to efficient cooling of low-entropy gas into galaxies.